How to select each of the segments in my bone for weight-paint?
Bone weighting only works on whole bones. Bone segments are still just part of a single bone, and are not individually selectable or even separate objects.

Blender will automatically blend each bone segment's transform over the vertices weighted to the whole bone, by proximity to the segment.

If you need to fine tune this more you need to use separate bones, and constrain them to move appropriately (mostly likely using a curve to control them in some way).